A fluid-structure interaction benchmark in turbulent flow (FSI-PfS-1a)

Sensitivity study for the subset case

In order to better understand the test case a comprehensive study on the influence of the three main parameters of the structure (the thickness of the plate $h$, the density $\rho_\text{rubber plate}$ and the Young's modulus $E$) was carried out.

The thickness of the plate was at first set to $h = 0.002$ m. However, as mentioned in Section~\ref{sec:Description_model}, the material is natural rubber and to manufacture a perfectly homogeneous 2 mm plate is not easy. The experimental measurements show that the thickness of the plate varies between 0.002 and 0.0022~m. Therefore, two values of $h$ are tested: the theoretical value of 0.002~m and the average value 0.0021~m.

The density of the plate $\rho_\text{rubber plate}$ is the second parameter. The value of $\rho_\text{rubber plate}$ is determined by a scale and the volume of the structure. Consequently, $\rho_\text{rubber plate}$ also depends on $h$. With $h = 0.002$ m $\rho_\text{rubber plate}$ is determined to be equal to 1425 kg m$^{-3}$. With $h = 0.0021$ m $\rho_\text{rubber plate}$ is found to be equal to 1360 kg m$^{-3}$.

The third parameter of the structure is the Young's modulus, because it has an important influence on the modeling of the material. A large spectrum of values for $E$ is tested to evaluate this influence.

All the tests were carried out without structural damping and are summarized in Table~\ref{Subset_case:parameters_study}. The full case used in Section~\ref{sec:Full_case_vs_Subset_case} and the experimental results are also added as references. Each simulation was done for a time interval of $4 $\,s physical time and comprises about 27 swiveling periods. The frequency $f_{FSI}$ of the swiveling mode and the extrema of the mean period of the FSI phenomenon (here the dimensionless y-displacement \mbox{$U_y^* = U_y / D$} as explained in Section~\ref{sec:Full_case_vs_Subset_case}) are compared. Furthermore, the relative errors between the numerical and experimental values is given.

Fig. xx Parameter study for the subset case of the FSI test case (without structural damping).

The following results and trends can be seen:

By varying the Young's modulus $E$ between 8 and 16 MPa it is possible to control the mode of the FSI phenomenon. Thus $E$ turns out to be the most crucial material parameter. With $E$ smaller than 9 MPa, the system oscillates in the second swiveling mode (similar to the second bending mode presented in Fig.~\ref{fig:eigenformen}). With $E$ larger than 12 MPa the structure deflection is dominated by the first bending mode of the structure. For a Young's modulus between 9 and 12 MPa a mode transition phase appears in which both swiveling modes are apparent. In this situation the y-displacements of the plate are no longer quasi-periodic and can not be described by a unique frequency.

Non-negligible variations in the density (1320 kg m$^{-3} \le \rho_\text{rubber plate} \le$ 1725 kg m$^{-3}$) for a fixed thickness (h = 0.002~m) and Young's modulus ($E$ = 14~MPa) do not drastically change the results of the frequency and of the mean period extrema. The FSI frequency $f_{FSI}$ slightly decreases with the increase of the density.

Comparing the results for both thicknesses for the range 14 $\le$ $E$ $\le$ 16~MPa, it is obvious that a mild variation of the thickness of the plate (0.1 mm, equivalent to 5 \%) has a non-negligible influence on the extrema of the mean period and no significant influence on the frequency.

Overall the frequency of the FSI phenomenon $f_{FSI}$ is very well predicted (relative error under 2.22 \%) for all tested parameters leading to the first swiveling mode.

Comparing the results for the density $\rho_\text{rubber plate}=1360 $ kg m$^{-3}$ in the range 14 $\le$ $E$ $\le$ 20~MPa, we observe that the FSI frequency $f_{FSI}$ slightly increases with the Young's modulus and that the displacement extrema decrease.

In summary, the parameter study shows that the Young's modulus is the most important parameter: It controls the swiveling mode of the plate. Furthermore, it can be observed that mild modifications of the shell thickness have a certain effect on the predicted FSI phenomenon. Contrarily, this parameter study shows that large variations of the density do not have major influence on the predictions. Therefore, errors in the density measurement does not play an important role. With the support of these extensive preliminary numerical investigations we can now compare the final numerical results with the experiment.

Comparison between numerical and experimental results

The investigations presented in Section~\ref{sec:Full_case_vs_Subset_case} based on slightly different material characteristics than defined in Section~\ref{sec:Material_parameters} have shown that the subset case permits a gain in CPU-time but nevertheless nearly identical results as the full case. Therefore, the numerical computation with the structural parameters defined in Section~\ref{sec:Material_parameters} ($E$ = 16 MPa, $h$ = 0.0021~m, $\rho_\text{rubber plate}$ = 1360 kg m$^{-3}$) is carried out for the subset case.

Two simulations are considered: one with the structural damping defined in Section~\ref{sec:validation_structure_model}, the other one without damping. These results are compared with the experimental data to check their accuracy. In order to quantitatively compare the experimental and numerical data, both are phase-averaged as explained in Section~\ref{sec:Generation_of_phase-resolved_data}. Similar to the numerical comparison presented in Section~\ref{sec:Full_case_vs_Subset_case} the displacement of the structure will be first analyzed and then the phase-resolved flow field is considered.

Fig. xx Experimental structural results: Structure contour for the reference period.

The structure contour of the phased-averaged experimental results for the reference period is depicted in Fig.~\ref{fig:swiveling_mode_FSI-PfS-1a}. Obviously, the diagram represents the first swiveling mode of the FSI phenomenon showing only one wave mode at the clamping. Figure~\ref{fig:comp_mean_period:a} depicts the experimental dimensionless raw signal obtained at a point located in the midplane at a distance of 9~mm from the shell extremity (see Fig.~\ref{fig:comp_full_subset:c}). Figure~\ref{fig:comp_mean_period:b} shows the numerical signal predicted without structural damping and Fig.~\ref{fig:comp_mean_period:c} the one computed with damping. Applying the phase-averaging process the mean phase of the FSI phenomenon for the experiment and for the simulations is generated. The outcome is presented in Fig.~\ref{fig:comp_mean_period:d} with the phase as the abscissa and the dimensionless displacement \mbox{$U_y^* = U_y / D$} as the ordinate. The amplitudes of the experimental signal varies more than in the predictions. Therefore, the maximal standard deviation of each point of the averaged phase is for the experiment bigger (0.083) than for the simulation (0.072 with and without damping). In order to check the reliability of the computed mean phase the coefficient of determination $R^2$ is computed: it is smaller for the mean experimental phase (0.9640) than for the mean simulation ones (0.9770 without damping and 0.9664 with damping). However, the values are close to unity, which is an indication that the averaged phases are representative for the signals. In Fig.~\ref{fig:comp_mean_period:d} the mean period calculated from the simulation without damping is quasi-antisymmetric with respect to \mbox{$U_y^* = 0$}. On the contrary the period derived from the experiment is not exactly antisymmetric with respect to the midpoint of the phase \mbox{$\phi =

 \pi$}: the cross-over is not at the midpoint of the phase but

slightly deviates to the right. However, the absolute values of the minimum and maximum are nearly identical. As for the experimental phase, the simulation with damping generates a phase signal, which is not completely antisymmetric. In the experiment this weak asymmetry can be attributed to minor asymmetries in the setup or in the rubber material. The comparison in Fig.~\ref{fig:comp_mean_period:d} shows some differences in the extrema and a summary is presented in Table~\ref{tab:comparison_num_exp_damping}. Without structural damping the simulations produce extrema which are too large by about 10~\%. With structural damping the extrema are smaller, even smaller than in the experiment by about 6~\%. Thus, the structural damping also has a significant influence on the FSI predictions and can not be overlooked.

The frequency of the FSI phenomenon, i.e., the frequency of the y-displacements, is about \mbox{$f_{{FSI}_{\text{exp}}}=7.10\,$Hz} in the experimental investigations, which corresponds to a Strouhal number \mbox{$\text{St} \approx 0.11$}. In the numerical predictions without damping this frequency is \mbox{$f_{{FSI}_{\text{num}}}^{\text{no damping}}=7.08\,$Hz} and with damping \mbox{$f_{{FSI}_{\text{num}}}^{\text{damping}}=7.18\,$Hz}. This comparison shows an error of \mbox{$\epsilon_{f}=-0.25\,\%$} for the results without damping and \mbox{$\epsilon_{f}=1.15\,\%$} for the cases with damping. Nevertheless, the FSI frequency is also very well predicted in both cases. One can notice that the frequency of the coupled system slightly increases due to the structural damping.

% \begin{table}[!htbp]

 \centering
 \begin{tabular}{|l||c|c|c|c|c|c|c|}
   \hline
   Case                   & \multicolumn{7}{|c|}{Results} \\
   \hline
                          & St (Hz) & $f_{FSI}$ (\%) & Error & \mbox{$\left.U_{y}^*\right|_{max}$} & Error (\%) & \mbox{$\left.U_{y}^*\right|_{min}$} & Error \\
   \hline
   \hline
   Sim. (no damping)      & 0.1125 & 7.08   & -0.25   & 0.456             & 9.1  & -0.464            & -10.6 \\
  \hline
   Sim. (damping)         & 0.1140 & 7.18   & 1.15    & 0.396             & -5.32 & -0.395            & 6.02   \\
   \hline
   \hline
   Experiments            & 0.1128 & 7.10   & -       & 0.418             & -     & -0.420            & -     \\
   \hline
 \end{tabular}
 \caption{\label{tab:comparison_num_exp_damping} 
   Comparison between numerical results with and without structural 
      damping and the experiment.}

\end{table}

Fig. xx Comparison of experimental and numerical results; raw signals and averaged phases of a point located at 9 mm distance from the shell extremity.

Phase-resolved flow field

Owing to improved results in case of the structural damping, this case is chosen for the direct comparison with the measurements. The phase-averaging process delivers the phase-resolved flow fields. Four phase-averaged positions, which describe the most important phases of the FSI phenomenon, are chosen for the comparison: Fig.~\ref{fig:comparison_rubber_plate:1} shows the flexible structure reaching a maximal upward deflection at \mbox{$t \approx T /

 4$}. Then, it deforms in the opposite direction and moves

downwards. At \mbox{$t \approx T / 2$} the shell is almost in its undeformed state (see Fig.~\ref{fig:comparison_rubber_plate:2}). Afterwards, the flexible structure reaches a maximal downward deformation at \mbox{$t \approx 3

 T / 4$} as seen in Fig.~\ref{fig:comparison_rubber_plate:3}. At

\mbox{$t \approx T$} the period cycle is completed and the shell is near its initial state presented in Fig.~\ref{fig:comparison_rubber_plate:4}.

For each of the given phase-averaged positions, the experimental and numerical results (dimensionless streamwise and transverse velocity component) are plotted for comparison. Note that the shell in the experimental figures is shorter than in the simulation plots. Indeed, in the experiment it is not possible to get exactly the whole experimental structure, around 1 mm at the end of the structure is missing. As in Section~\ref{sec:Comparison_of_numerical_results} an additional figure shows the error between the simulation and the experiment for the velocity magnitude.

At \mbox{$t \approx T / 4$} (see Fig.~\ref{fig:comparison_rubber_plate:1}), when the structure is in its maximal upward deflection, the acceleration zone above the structure has reached its maximum. The acceleration area below the plate is growing. Both phenomena are correctly predicted in the simulations. The computed acceleration area above the structure is slightly overestimated. However the local error is mostly under 20 \%.

The separation points at the cylinder are found to be in close agreement between measurements and predictions. Accordingsly, also the location of the shear layers shows a good agreement between simulations and experiments. The shedding phenomenon behind the structure generates a turbulent wake, which is correctly reproduced by the computations. Owing to the phase-averaging procedure, as expected all small-scale structures are averaged out.

At \mbox{$t \approx T / 2$} (see Fig.~\ref{fig:comparison_rubber_plate:2}), the plate is near its undeformed state. The acceleration zone above the structure has shrunk in favor of the area below the plate. Regarding these areas the predictions show a very good agreement with the measurements (marginal local errors). The predicted wake directly behind the structure matches the measured one.

At \mbox{$t \approx 3 T / 4$} (see Fig.~\ref{fig:comparison_rubber_plate:3}), the downward deformation of the plate is maximal, the flow is the symmetrical to the flow observed at \mbox{$t \approx T / 4$} with respect to \mbox{y/D = 0}. Again the acceleration areas around the structure show a very good agreement with the measurements. Once more the wake is correctly predicted in the near-field of the structure.

At \mbox{$t \approx T$} (see Fig.~\ref{fig:comparison_rubber_plate:4}) the flow is symmetrical to the flow observed at \mbox{$t \approx T /

 2$} with respect to \mbox{y/D = 0}. The computed acceleration area

above the structure is slightly overestimated, but the local error is under 20 \%. The wake is again correctly predicted except directly after the flexible structure.

For every position the local error is mostly under $20\,\%$. In the error plot the areas with a bigger local error are near the structure and in the shear layers. This can be explained by the fact that near the structure and in the shear layers the gradients of the flow quantities are large. Since the mesh used for the simulation is much finer than the PIV measurement grid, the accuracy of the numerical solution is much higher than the precision of the PIV measurements in these regions. Another reason is that the error expected by the PIV method is more important for low flow velocities. Close to the flexible structure and directly after its tail the flow velocity is small, which at least partially explains the deviations observed between the experimental and numerical results.

In summary, for every position the computed flow is in good agreement with the measured one. The shedding phenomenon behind the cylinder and the positions of the vortices convected downstream are correctly predicted.

Contributed by: Michael Breuer — Helmut-Schmidt Universität Hamburg